Some questions on the univalence of functions of the class Σ

In this paper we give an example of two convex functions in ¦ ζ¦ > 1 whose arithmetic mean is nonconvex. We calculate the radius of convexity of the sum of two convex functions; it is equal to \(\sqrt {1 + \sqrt 2 } \) . For functions F(ζ)=ζ+b₁/ζ+..., where F′(ζ)=f(ζ)/ζ, iff(ζ) = ζ + a₁/ζ+... is univalent ¦ζ¦ > 1, then the radius of univalence is the root of the equation 4E· (1/r)/K(1/r)+1/r²=3.     

Another proof of Grothendieck’s theorem on the splitting of vector bundles on the projective line

This note contains another proof of Grothendieck‘s theorem on the splitting of vector bundles on the projective line over a field k. Actually the proof is formulated entirely in the classical terms of a lattice \(\Lambda \cong k[T]^d\), discretely embedded into the vector space \(V \cong K_\infty ^d\), where \(K_\infty \cong k((1/T))\) is the completion of the field of rational functions k(T) at the place \(\infty \) with the usual valuation.     

A Note on Self-Mapping of the Interval

In this note we showed by construct a example that for, the con ditions and are not equivalent. Let, define by, for any We can construct a example as following: Let be identity, define by     

Some Properties of the Metahomomorpisms on Groups

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Note on the Paper “A Note on the SVD of Matrices”

Let F be a field, R be an F-algebra having SVD property['], R+ = {a e Rla = a}. In[2], we proved thatLemma 1 Let R be an F-algebra which having SVD proPerty Then R+ is a sub6e1d inF, R+ is a fOrmaIly rea1 6eId, and R+ is Pythagorean with no zerChdivisor.In the proof of Lernma 1 of [2], thereis this step: "FOr any a E R+ l clearly, there ekistsd E R+ n F such that a2 = ad = a', thus a = la E F, is fOllOws that R+ C F".Recently, paper [11 shows that "The proof of this step is no…     

New results on the geometry of the moduli space of Riemann surfaces

We briefly survey our recent results about the Mumford goodness of several canonical metrics on the moduli spaces of Riemann surfaces,including the Weil-Petersson metric,the Ricci metric, the Perturbed Ricci metric and the Kahler-Einstein metric.We prove the dual Nakano negativity of the Weil-Petersson metric.As applications of these results we deduce certain important results about the L~2-cohomology groups of the logarithmic tangent bundle over the compactifled moduli spaces.     

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.     

Study on the Structure of Γ-Semihypergroups

In this article, we study Γ-semihypergroups introduced recently. We introduce the hyper versions of Green's relations in Γ-semihypergroups and give some related characterizations. We introduce the n-prime Γ-hyperideal and n-semiprime Γ-hyperideal of a Γ-semihypergroup and show that for any integer n ≥ 2, n-prime Γ-hyperideals are a generalization of prime Γ-ideals. We also give the relationship between n-prime Γ-hyperideals and Γ-hyperideal extensions in Γ-semihypergroups. Then, we deal with prime ideal and introduce the notion of prime radical in a Γ-semihypergroup M, and also, we obtain some results and relations among the prime radicals of M and S, where S is the left operator semihypergroup of M.     

Extremal problems on the Hua domain of the first type

In this paper we study some extremal problems between the Hua domain of the first type and the unit ball.     

Equivalence Relation on the Set of Subsets of Z_v and Enumeration of the Equivalence Classes

Motivated by the concept of equivalent cyclic difference sets on Z_v (the residue ring modulo v), we introduce a similar equivalence relation on the set of subsets of Z_v as follows. Difinition Let D_1 and D_2 be two subsets of Z_v. If D_1=tD_2+s for some t, s∈Z_v(ged(t,v)=1), then D_1 and D_2 are said to be equivalent, denoted by D_～D_2.     

On the Monotonicity of the Speed of Random Walks on a Percolation Cluster of Trees

The authors consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.     

Note on the Norm Involved in the Definition of the ε-spectrum

There exist many definitions of the ε-spectrum. Unlike the spectrum, the ε-spectrum definitions depend on the choice of the norm. We propose to study the conditions on the norm that make all these definitions equivalent. The spectral radius of a related matrix is a useful tool in the eigenvalues perturbation analysis. The study of its connection with the ε-spectrum provides a new definition of the ε-spectrum. In most articles dealing with this subject, the chosen norm is the spectral one. In this paper, we intend to justify this choice.     

On the Convergence of Generalizations of the Sinc Approximations on the Privalov–Chanturia Class

Journal of Applied and Industrial Mathematics - We establish some condition in terms of the left or right modulus of continuity and the negative or positive modulus of variation of a function $$f...     

Some Results on Evolutionary Equations Involving Functions of the Eigenvalues of the Hessian

How to design the parabolic counterpart of the nonlinear elliptic equation involving function of the eigenvalues of the Hessian is discussed and the first initial-boundary value problem for such a parabolic equation is studied.     

On the Boundary of the Set of Morse-Smale Vector Fields on Two-dimensional Manifolds

NotatianM a compact,boundaryless,orientable or non—orientable two-dimensional manifold of class C~∞H~r(M) the Banach space of C~r(r≥1)vector fields on M∑~r(M) the set of structurally stable vector fields in H~r(M)MS~r(M) the set of Morse—Smale vector fields in H~r(M)     

Improving the Clark–Suen bound on the domination number of the Cartesian product of graphs

A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, $\gamma (G\square H)\ge \gamma \left(G\right)\gamma \left(H\right)∕2$, where $\gamma$ stands for the domination number, and $G\square H$ is the Cartesian product of graphs $G$ and $H$. In this note, we improve this bound by employing the 2-packing number $\rho \left(G\right)$ of a graph $G$ into the formula, asserting that $\gamma (G\square H)\ge (2\gamma \left(G\right)?\rho \left(G\right))\gamma \left(H\right)∕3$. The resulting bound is better than that of Clark and Suen whenever $G$ is a graph with $\rho \left(G\right)<\gamma \left(G\right)∕2$, and in the case $G$ has diameter 2 reads as $\gamma (G\square H)\ge (2\gamma \left(G\right)?1)\gamma \left(H\right)∕3$.     

Randomization of Q_p spaces on the unit ball of C~n

In this paper, we study the random power series of several complex variables, and give some sufficient conditions for random power series to belong to Qp and Qp,0 spaces by means of their characterizations of Carleson measure. Moreover, we show that such conditions are best possible in some sense.